Integrand size = 20, antiderivative size = 68 \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1128, 654, 635, 212} \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
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Rule 212
Rule 635
Rule 654
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right ) \\ & = \frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 c} \\ & = \frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x^2}{\sqrt {a+b x^2+c x^4}}\right )}{2 c} \\ & = \frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82
method | result | size |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\) | \(56\) |
risch | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\) | \(56\) |
elliptic | \(\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}-\frac {b \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\) | \(56\) |
pseudoelliptic | \(\frac {b \ln \left (2\right )-b \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{4 c^{\frac {3}{2}}}\) | \(65\) |
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Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.37 \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\left [\frac {b \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, \sqrt {c x^{4} + b x^{2} + a} c}{8 \, c^{2}}, \frac {b \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} c}{4 \, c^{2}}\right ] \]
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\[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {x^{3}}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {b \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{4 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2} + a}}{2 \, c} \]
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Time = 13.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {c\,x^4+b\,x^2+a}}{2\,c}-\frac {b\,\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )}{4\,c^{3/2}} \]
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